Possible interpretation: ‘marriage market’ Two populations, men and women; matching: one individual from each population We want to explain matching patterns (who marries whom): assortative matching (by education, income,...); impact on inequality, etc. ... but also: how are the gain from marriage allocated ? ... and: how does the market for marriage a¤ects individual and household behavior: ex ante : human capital investment of future spouses (basic idea: HC improves marital prospects, in many directions) ex post : human capital investment of existing couples (basic idea: expenditures may depend on the spouses’ respective ‘powers’ - cf collective model). ‘Tractable General Equilibrium’ Di¤erent models are better suited for some purposes than for others. P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 3 / 76
Issues related to matching: two examples P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 4 / 76
Example 1: Assortative matching and inequality Burtless (EER 1999): over 1979-1996, ‘The changing correlation of husband and wife earnings has tended to reinforce the e¤ect of greater pay disparity.’ P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 5 / 76
Example 1: Assortative matching and inequality Burtless (EER 1999): over 1979-1996, ‘The changing correlation of husband and wife earnings has tended to reinforce the e¤ect of greater pay disparity.’ Maybe 1/3 of the increase in household-level inequality (Gini) comes from rise of single-adult households and 1/6 from increased assortative matching. P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 5 / 76
Example 1: Assortative matching and inequality Burtless (EER 1999): over 1979-1996, ‘The changing correlation of husband and wife earnings has tended to reinforce the e¤ect of greater pay disparity.’ Maybe 1/3 of the increase in household-level inequality (Gini) comes from rise of single-adult households and 1/6 from increased assortative matching. Several questions; in particular: P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 5 / 76
Example 1: Assortative matching and inequality Burtless (EER 1999): over 1979-1996, ‘The changing correlation of husband and wife earnings has tended to reinforce the e¤ect of greater pay disparity.’ Maybe 1/3 of the increase in household-level inequality (Gini) comes from rise of single-adult households and 1/6 from increased assortative matching. Several questions; in particular: Why did correlation change? Did ‘preferences for assortativeness’ change? P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 5 / 76
Example 1: Assortative matching and inequality Burtless (EER 1999): over 1979-1996, ‘The changing correlation of husband and wife earnings has tended to reinforce the e¤ect of greater pay disparity.’ Maybe 1/3 of the increase in household-level inequality (Gini) comes from rise of single-adult households and 1/6 from increased assortative matching. Several questions; in particular: Why did correlation change? Did ‘preferences for assortativeness’ change? How do we compare single-adult households and couples? What about intrahousehold inequality? P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 5 / 76
Example 2: College premium and the demand for college education Motivation: remarkable increase in female education, labor supply, incomes worldwide during the last decades. Source: Becker-Hubbard-Murphy 2009 P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 6 / 76
Example 2: College premium and the demand for college education In the US: P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 7 / 76
Example 2: College premium and the demand for college education Questions : why such di¤erent responses by gender? P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 8 / 76
Example 2: College premium and the demand for college education Questions : why such di¤erent responses by gender? impact on intrahousehold allocation? P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 8 / 76
Example 2: College premium and the demand for college education Questions : why such di¤erent responses by gender? impact on intrahousehold allocation? impact on household behavior (expenditure, HC investment, etc.) ! especially relevant in developing countries! P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 8 / 76
Roadmap Matching models: general presentation 1 The case of Transferable Utility (TU) 2 Extensions: 3 Pre-investment Multidimensional matching Imperfectly Transferable Utility Risk sharing Econometric implementation 4 P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 9 / 76
Roadmap Matching models: general presentation 1 The case of Transferable Utility (TU) 2 Extensions: 3 Pre-investment Multidimensional matching Imperfectly Transferable Utility Risk sharing Econometric implementation 4 P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 10 / 76
Matching models: three main families Matching under NTU (Gale-Shapley) 1 Idea: no transfer possible between matched partners Matching under TU (Becker-Shapley-Shubik) 2 Transfers possible without restrictions Technology: constant ‘exchange rate’ between utiles In particular: (strong) version of interpersonal comparison of utilities ! requires restrictions on preferences Matching under Imperfectly TU (ITU) 3 Transfers possible But no restriction on preferences ! technology involves variable ‘exchange rate’ ... plus ‘general’ approaches (’matching with contracts’, from Kelso-Crawford to Milgrom-Hat…eld-Kominers and friends) ... and links with: auction theory, general equilibrium. P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 11 / 76
U h U w Pareto frontier: NTU
U h U w Pareto frontier: TU
U h U w Pareto frontier: ITU
Matching models: three main families Similarities and di¤erences All aimed at understanding who is matched with whom Only the last 2 address how the surplus is divided Only the third allows for impact on the group’s aggregate behavior P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 12 / 76
Formal structure: Common components Compact, separable metric spaces X , Y (‘women, men’) with …nite measures F and G . Note that the spaces may be multidimensional Spaces X , Y often ‘completed’ to allow for singles: X = X [ f ∅ g , ¯ ¯ Y = Y [ f ∅ g A matching de…nes of a measure h on X � Y (or ¯ X � ¯ Y ) such that the marginals of h are F and G The matching is pure if the support of the measure is included in the graph of some function φ Translation: matching is pure if y = φ ( x ) a.e. ! no ‘randomization’ P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 13 / 76
Formal structure: di¤erences De…ning the problem : populations X , Y plus P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76
Formal structure: di¤erences De…ning the problem : populations X , Y plus NTU: two funtions u ( x , y ) , v ( x , y ) P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76
Formal structure: di¤erences De…ning the problem : populations X , Y plus NTU: two funtions u ( x , y ) , v ( x , y ) TU: one function s ( x , y ) (intrapair allocation is endogenous) P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76
Formal structure: di¤erences De…ning the problem : populations X , Y plus NTU: two funtions u ( x , y ) , v ( x , y ) TU: one function s ( x , y ) (intrapair allocation is endogenous) ITU: Pareto frontier u = F ( x , y , v ) P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76
Formal structure: di¤erences De…ning the problem : populations X , Y plus NTU: two funtions u ( x , y ) , v ( x , y ) TU: one function s ( x , y ) (intrapair allocation is endogenous) ITU: Pareto frontier u = F ( x , y , v ) De…ning the solution P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76
Formal structure: di¤erences De…ning the problem : populations X , Y plus NTU: two funtions u ( x , y ) , v ( x , y ) TU: one function s ( x , y ) (intrapair allocation is endogenous) ITU: Pareto frontier u = F ( x , y , v ) De…ning the solution NTU: only the measure h ; stability as usual P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76
Formal structure: di¤erences De…ning the problem : populations X , Y plus NTU: two funtions u ( x , y ) , v ( x , y ) TU: one function s ( x , y ) (intrapair allocation is endogenous) ITU: Pareto frontier u = F ( x , y , v ) De…ning the solution NTU: only the measure h ; stability as usual TU: measure h and two functions u ( x ) , v ( y ) such that u ( x ) + v ( y ) = s ( x , y ) for ( x , y ) 2 Supp ( h ) and stability u ( x ) + v ( y ) � s ( x , y ) for all ( x , y ) P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76
Formal structure: di¤erences De…ning the problem : populations X , Y plus NTU: two funtions u ( x , y ) , v ( x , y ) TU: one function s ( x , y ) (intrapair allocation is endogenous) ITU: Pareto frontier u = F ( x , y , v ) De…ning the solution NTU: only the measure h ; stability as usual TU: measure h and two functions u ( x ) , v ( y ) such that u ( x ) + v ( y ) = s ( x , y ) for ( x , y ) 2 Supp ( h ) and stability u ( x ) + v ( y ) � s ( x , y ) for all ( x , y ) ITU: measure h and two functions u ( x ) , v ( y ) such that u ( x ) = F ( x , y , v ( y )) for ( x , y ) 2 Supp ( h ) and stability u ( x ) � F ( x , y , v ( y )) for all ( x , y ) P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 14 / 76
Formal structure: di¤erences (cont.) Characterization: P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76
Formal structure: di¤erences (cont.) Characterization: NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76
Formal structure: di¤erences (cont.) Characterization: NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76
Formal structure: di¤erences (cont.) Characterization: NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76
Formal structure: di¤erences (cont.) Characterization: NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c Stability equivalent to surplus maximization P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76
Formal structure: di¤erences (cont.) Characterization: NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c Stability equivalent to surplus maximization therefore: existence easy to establish P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76
Formal structure: di¤erences (cont.) Characterization: NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c Stability equivalent to surplus maximization therefore: existence easy to establish ‘generic’ uniqueness P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76
Formal structure: di¤erences (cont.) Characterization: NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c Stability equivalent to surplus maximization therefore: existence easy to establish ‘generic’ uniqueness In a nutshell P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76
Formal structure: di¤erences (cont.) Characterization: NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c Stability equivalent to surplus maximization therefore: existence easy to establish ‘generic’ uniqueness In a nutshell NTU: intragroup allocation exogenously imposed ; transfers are ruled out by assumption P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76
Formal structure: di¤erences (cont.) Characterization: NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c Stability equivalent to surplus maximization therefore: existence easy to establish ‘generic’ uniqueness In a nutshell NTU: intragroup allocation exogenously imposed ; transfers are ruled out by assumption TU and ITU: intragroup allocation endogenous ; transfers are paramount and determined (or constrained) by equilibrium conditions P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76
Formal structure: di¤erences (cont.) Characterization: NTU: existence (Gale-Shapley), uniqueness not guaranteed (lattice structure of the set of stable matchings) ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley), uniqueness not guaranteed TU: highly speci…c Stability equivalent to surplus maximization therefore: existence easy to establish ‘generic’ uniqueness In a nutshell NTU: intragroup allocation exogenously imposed ; transfers are ruled out by assumption TU and ITU: intragroup allocation endogenous ; transfers are paramount and determined (or constrained) by equilibrium conditions TU: life much easier (GQL ! equivalent to surplus maximization) ... ... but price to pay: couple’s (aggregate) behavior does not depend on ‘powers’, therefore on equilibrium conditions P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 15 / 76
Implications (crucial for empirical implementation) NTU: stable matchings solve u ( x ) = max z f U ( x , z ) j V ( x , z ) � v ( z ) g and v ( y ) = max z f V ( z , y ) j U ( z , y ) � u ( z ) g for some pair of functions u and v . P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 16 / 76
Implications (crucial for empirical implementation) NTU: stable matchings solve u ( x ) = max z f U ( x , z ) j V ( x , z ) � v ( z ) g and v ( y ) = max z f V ( z , y ) j U ( z , y ) � u ( z ) g for some pair of functions u and v . TU: stable matchings solve u ( x ) = max z f s ( x , z ) � v ( z ) g and v ( y ) = max z f s ( z , y ) � u ( z ) g for some pair of functions u and v . P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 16 / 76
Implications (crucial for empirical implementation) NTU: stable matchings solve u ( x ) = max z f U ( x , z ) j V ( x , z ) � v ( z ) g and v ( y ) = max z f V ( z , y ) j U ( z , y ) � u ( z ) g for some pair of functions u and v . TU: stable matchings solve u ( x ) = max z f s ( x , z ) � v ( z ) g and v ( y ) = max z f s ( z , y ) � u ( z ) g for some pair of functions u and v . ITU: stable matchings solve z f F � 1 ( z , y , u ( z )) g u ( x ) = max z f F ( x , z , v ( z )) g and v ( y ) = max for some pair of functions u and v . P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 16 / 76
Roadmap Matching models: general presentation 1 The case of Transferable Utility (TU) 2 Extensions: 3 Pre-investment Multidimensional matching Imperfectly Transferable Utility Risk sharing Econometric implementation 4 P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 17 / 76
Transferable Utility (TU) De…nition A group satis…es TU if there exists monotone transformations of individual utilities such that the Pareto frontier is an hyperplane u ( x ) + v ( y ) = s ( x , y ) for all values of prices and income. ! Marriage market: assumption on preferences? Model: collective (public and private consumptions, e¢cient decisions) TU if ‘Generalized Quasi Linear (GQL, Bergstrom and Cornes 1981): � � � + q 1 � q 2 i , ..., q n u i ( q i , Q ) = F i i b i ( Q ) A i i , Q with b i ( Q ) = b ( Q ) for all i (much more general than QL) Then standard model: x , y incomes and: s ( x , y ) = H ( x + y ) = max F � 1 ( u 1 ) + F � 1 ( u 2 ) under BC 1 2 P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 18 / 76
Basic result If a matching is stable, the corresponding measure satis…es the surplus maximization problem , which is an optimal transportation problem (Monge-Kantorovitch): Find a measure h on X � Y such that the marginals of h are F and G , and h solves Z max X � Y s ( x , y ) dh ( x , y ) h Hence: linear programming P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 19 / 76
Basic result If a matching is stable, the corresponding measure satis…es the surplus maximization problem , which is an optimal transportation problem (Monge-Kantorovitch): Find a measure h on X � Y such that the marginals of h are F and G , and h solves Z max X � Y s ( x , y ) dh ( x , y ) h Hence: linear programming Dual problem: dual functions u ( x ) , v ( y ) and solve Z Z min X u ( x ) dF ( x ) + Y v ( y ) dG ( y ) u , v under the constraint u ( x ) + v ( y ) � s ( x , y ) for all ( x , y ) 2 X � Y P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 19 / 76
Basic result If a matching is stable, the corresponding measure satis…es the surplus maximization problem , which is an optimal transportation problem (Monge-Kantorovitch): Find a measure h on X � Y such that the marginals of h are F and G , and h solves Z max X � Y s ( x , y ) dh ( x , y ) h Hence: linear programming Dual problem: dual functions u ( x ) , v ( y ) and solve Z Z min X u ( x ) dF ( x ) + Y v ( y ) dG ( y ) u , v under the constraint u ( x ) + v ( y ) � s ( x , y ) for all ( x , y ) 2 X � Y In particular, the dual variables u and v describe an intrapair allocation compatible with a stable matching P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 19 / 76
Links with hedonic models Structure: three sets (‘buyers’ X , ‘sellers’ Y , ‘products’ Z ) with measures µ , ν , σ . B Buyer x : quasi linear preferences U ( x , z ) � P ( z ) ; seller y maximizes pro…t P ( z ) � c ( y , z ) Equilibrium: price function P ( z ) that clear markets Technically: function P and measure α on the product set X � Y � Z such that (i) marginal of α on X (resp. Y ) coincides with µ (resp. ν ) (ii) for all ( x , y , z ) in the support of α , � � x , z 0 � � P � z 0 �� U ( x , z ) � P ( z ) = max U z 0 2 K � � z 0 � � c � y , z 0 �� and P ( z ) � c ( y , z ) = max P . z 0 2 K P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 20 / 76
Links with hedonic models Chiappori, McCann and Nesheim (2010): canonical correspondance between QL hedonic models and matching models under TU. Speci…cally: Consider a hedonic model and de…ne surplus: s ( x , y ) = max z 2 Z ( U ( x , z ) � c ( y , z )) Let η be the marginal of α over X � Y , u ( x ) and v ( y ) by u ( x ) = max z 2 K U ( x , z ) � P ( z ) and v ( y ) = max z 2 K P ( z ) � c ( y , z ) Then ( η , u , v ) de…nes a stable matching Conversely, starting from a stable matching ( η , u , v ) , u ( x ) + v ( y ) � s ( x , y ) � U ( x , z ) � c ( y , z ) ) c ( y , z ) + v ( y ) � U ( x , z For any z , take P ( z ) such that y 2 J f c ( y , z ) + v ( y ) g � P ( z ) � sup inf f u ( x , z ) � u ( x ) g x 2 I then P ( z ) is an equilibrium price for the hedonic model. P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 21 / 76
Supermodularity and assortative matching Assume X , Y one-dimensional. Then s is supermodular if whenever x > x 0 and y > y 0 then � x 0 , y 0 � > s � x , y 0 � + s � � x 0 , y s ( x , y ) + s Interpretation: single crossing (Spence - Mirrlees) Consequence: matching is assortative Generalization (CMcCN ET 2010): De…nition A surplus function s : X � Y � ! [ 0 , ∞ [ is said to be X � twisted if there is a set X L � X 0 of zero volume such that ∂ x s ( x 0 , y 1 ) is disjoint from ∂ x s ( x 0 , y 2 ) for all x 0 2 X 0 n X L and y 1 6 = y 2 in Y . Then the stable matching is unique and pure P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 22 / 76
Intracouple allocation under TU Discrete number of agents: equilibrium (stability) conditions impose constraints on individual shares... P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 23 / 76
Intracouple allocation under TU Discrete number of agents: equilibrium (stability) conditions impose constraints on individual shares... ... but there exists in general an in…nite set of intramatch allocations P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 23 / 76
Intracouple allocation under TU Discrete number of agents: equilibrium (stability) conditions impose constraints on individual shares... ... but there exists in general an in…nite set of intramatch allocations However, basic result: With a continuum of agents, intramatch allocation of welfare is pinned down by the equilibrium conditions P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 23 / 76
Intracouple allocation under TU Discrete number of agents: equilibrium (stability) conditions impose constraints on individual shares... ... but there exists in general an in…nite set of intramatch allocations However, basic result: With a continuum of agents, intramatch allocation of welfare is pinned down by the equilibrium conditions Known from the outset, but ... P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 23 / 76
Intracouple allocation under TU Discrete number of agents: equilibrium (stability) conditions impose constraints on individual shares... ... but there exists in general an in…nite set of intramatch allocations However, basic result: With a continuum of agents, intramatch allocation of welfare is pinned down by the equilibrium conditions Known from the outset, but ... ... much easier than you would think P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 23 / 76
Pinning down intracouple allocation under TU Assume X , Y one dimensional and s supermodular. Then 3 steps Step 1: supermodularity implies assortative matching: x matched with y = ψ ( x ) if the number of women above x equals the number of men above ψ ( x ) P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 24 / 76
Pinning down intracouple allocation under TU Assume X , Y one dimensional and s supermodular. Then 3 steps Step 1: supermodularity implies assortative matching: x matched with y = ψ ( x ) if the number of women above x equals the number of men above ψ ( x ) Step 2: Stability implies u ( x ) = max s ( x , y ) � v ( y ) y with the max being reached for y = ψ ( x ) . Therefore u 0 ( x ) = ∂ s ∂ x ( x , ψ ( x )) and v 0 ( y ) = ∂ s ∂ y ( φ ( y ) , y ) and Z x Z y ∂ s ∂ s ∂ x ( t , ψ ( t )) dt , v ( y ) = k 0 + u ( x ) = k + ∂ y ( φ ( s ) , s ) ds 0 0 ! Utilities de…ned up to two additive constants P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 24 / 76
Pinning down intracouple allocation under TU Step 3: pin down the constants P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 25 / 76
Pinning down intracouple allocation under TU Step 3: pin down the constants Note that u ( x ) + v ( ψ ( x )) = s ( x , ψ ( x )) which pins down the sum k + k 0 P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 25 / 76
Pinning down intracouple allocation under TU Step 3: pin down the constants Note that u ( x ) + v ( ψ ( x )) = s ( x , ψ ( x )) which pins down the sum k + k 0 If one gender in excess supply (say women): the ‘last married’ woman indi¤erent between marriage and singlehood P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 25 / 76
Pinning down intracouple allocation under TU Step 3: pin down the constants Note that u ( x ) + v ( ψ ( x )) = s ( x , ψ ( x )) which pins down the sum k + k 0 If one gender in excess supply (say women): the ‘last married’ woman indi¤erent between marriage and singlehood Note: typically, discontinuity P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 25 / 76
Pinning down intracouple allocation under TU Step 3: pin down the constants Note that u ( x ) + v ( ψ ( x )) = s ( x , ψ ( x )) which pins down the sum k + k 0 If one gender in excess supply (say women): the ‘last married’ woman indi¤erent between marriage and singlehood Note: typically, discontinuity If equal number (knife-edge situation), indeterminate ... ... unless corner solutions P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 25 / 76
Applications: applied theory Various applications: Abortion and female empowerment (CO JPE 2006) Children and divorce (CW JoLE 2007) Male and female demand for higher education (CIW AER 2009) Dynamics: divorce and impact of divorce laws (CIW 10) Multidimensional matching: general framework (Galichon Salanié 2011) income/education and physical attractiveness (COQ 2011) income and smoking habits (COQ 2012) income and ‘reproductive capital’ (Low 2012) P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 26 / 76
Roadmap Matching models: general presentation 1 The case of Transferable Utility (TU) 2 Extensions 3 Pre-investment Multidimensional matching Imperfectly Transferable Utility Risk sharing Econometric implementation 4 P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 27 / 76
Extensions: roadmap Pre-investment 1 Multidimensional matching 2 Theory Practical Implementation ITU 3 General presentation A speci…c model Risk sharing 4 P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 28 / 76
Extensions: roadmap Pre-investment 1 Multidimensional matching 2 Theory Practical Implementation ITU 3 General presentation A speci…c model Risk sharing 4 P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 29 / 76
Pre-investment Two stage game: Agents independently (non cooperatively) invest in characteristics 1 (say in HC) Agents match on these characteristics 2 Model solved backwards: For given distributions of characteristics, matching equilibrium pins down the allocation of the surplus This allocation de…nes the return from the …rst period investment ’Rational expectations’: the distribution of characteristics expected by the agents when investing is realized by their investment P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 30 / 76
Is pre-investment e¢cient? Two opposite arguments: (‘Free rider’): My investment will increase the joint surplus, some of 1 which goes to my (future) partner ! under investment P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 31 / 76
Is pre-investment e¢cient? Two opposite arguments: (‘Free rider’): My investment will increase the joint surplus, some of 1 which goes to my (future) partner ! under investment (’Rat race’): I am competing again other potential spouses, I have to 2 be better ! over investment P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 31 / 76
Is pre-investment e¢cient? Two opposite arguments: (‘Free rider’): My investment will increase the joint surplus, some of 1 which goes to my (future) partner ! under investment (’Rat race’): I am competing again other potential spouses, I have to 2 be better ! over investment In fact: the investment is e¢cient 3 Why? u 0 ( x ) = ∂ s ∂ x ( x , ψ ( x )) P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 31 / 76
Is pre-investment e¢cient? Two opposite arguments: (‘Free rider’): My investment will increase the joint surplus, some of 1 which goes to my (future) partner ! under investment (’Rat race’): I am competing again other potential spouses, I have to 2 be better ! over investment In fact: the investment is e¢cient 3 Why? u 0 ( x ) = ∂ s ∂ x ( x , ψ ( x )) Application: gender unbalance: who invests more? (ACM) 4 P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 31 / 76
Extensions: roadmap Pre-investment 1 Multidimensional matching 2 Theory Practical Implementation ITU 3 General presentation A speci…c model Risk sharing 4 P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 32 / 76
Two-dimensional example: X � R 2 , Y � R 2 Surplus S ( x 1 , x 2 , y 1 , y 2 ) Particular case (‘index’): S ( x 1 , x 2 , y 1 , y 2 ) = S ( A ( x 1 , x 2 ) , B ( y 1 , y 2 )) T wo questions: Who marries whom? 1 How is the surplus shared? 2 P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 33 / 76
Who marries whom? Two possible approaches: ‘Guess’ what the matching patterns will look like; then: 1 Compute the thresholds Compute the individual utilities (see below) Check the stability conditions Use surplus maximization 2 Always possible Typically: optimal control Very useful for simulations, etc. Common caveat: matching may not be ‘pure’ P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 34 / 76
Purity Idea: generalize the one-dimensional ‘supermodularity ) assortativeness’ result Generalization of supermodularity (CMcCN ET 2010): De…nition A surplus function S : X � Y � ! [ 0 , ∞ [ is said to be X � twisted if there is a set X L � X 0 of zero volume such that ∂ x S ( x 0 , y 1 ) is disjoint from ∂ x S ( x 0 , y 2 ) for all x 0 2 X 0 n X L and y 1 6 = y 2 in Y . Then the stable matching is unique and pure De…nition The matching is pure if the measure h is born by the graph of a function: for almost all x there exists exactly one y such that x matched with y . If not: ‘randomization’: an open set of (say) women are indi¤erent between several men P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 35 / 76
Who marries whom? (cont.) Assume the condition is satis…ed: ( y 1 , y 2 ) = φ ( x 1 , x 2 ) . Then surplus maximization: Z max X S ( x 1 , x 2 , φ ( x 1 , x 2 )) dF ( x 1 , x 2 ) φ with a constraint: The push-forward of F through φ coincides with G where the push-forward φ # F of F through φ de…ned by � � φ � 1 ( B ) φ # F ( B ) = F for any Borel B � X ! Optimal control P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 36 / 76
Sharing the surplus As previously, 3 steps Step 1: ( x 1 , x 2 ) matched with ( y 1 , y 2 ) = ψ ( x 1 , x 2 ) P.A. Chiappori (Columbia University) Matching with Transfers IIES, Stockholm, May 2013 37 / 76
Recommend
More recommend
